After the progress in a decade, graphene, or the nanotube and nanoribbon,
has become one of the most active two-dimensional nanomaterial in condensed
matter physics due to excellent electrical and mechanical properties [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}. The monolayer
graphene (MLG) is a zero-gap semiconductor where the conduction and valance
band touch at the Dirac points [8]}, [9]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}. It has a trivial bulk topology as
two inequivalent valleys provide opposite topological charges, leading to a
zero Chern number [18]}. However, edge states still exist in such
graphene systems as non-dispersive bands (flat bands) at the Fermi level
[8]}, [9]}, [21]}, [22]}, [23]}, [24]}, which are observed by supposing a
semi-infinite system, with quantized wave vectors \(k_{y}\)  in the infinite
direction [25]}, [21]}, [23]}, [24]}, [29]}. The existence of these non-dispersive
edge states and related topology in the MLG can be further described by the
bulk-edge correspondence between winding number or Zak phase of
one-dimensional (1D) SSH chain systems and the existence of a localized
state at the chain's edge [25]}, [21]}, [24]}, as shown in Fig. REF .
